Asymptotic nonparametric statistical analysis of stationary time series

Stationarity is a very general, qualitative assumption, that can be assessed on the basis of application specifics. It is thus a rather attractive assumption to base statistical analysis on, especially for problems for which less general qualitative assumptions, such as independence or finite memory, clearly fail. However, it has long been considered too general to allow for statistical inference to be made. One of the reasons for this is that rates of convergence, even of frequencies to the mean, are not available under this assumption alone. Recently, it has been shown that, while some natural and simple problems such as homogeneity, are indeed provably impossible to solve if one only assumes that the data is stationary (or stationary ergodic), many others can be solved using rather simple and intuitive algorithms. The latter problems include clustering and change point estimation. In this volume I summarize these results. The emphasis is on asymptotic consistency, since this the strongest property one can obtain assuming stationarity alone. While for most of the problems for which a solution is found this solution is algorithmically realizable, the main objective in this area of research, the objective which is only partially attained, is to understand what is possible and what is not possible to do for stationary time series. The considered problems include homogeneity testing, clustering with respect to distribution, clustering with respect to independence, change-point estimation, identity testing, and the general question of composite hypotheses testing. For the latter problem, a topological criterion for the existence of a consistent test is presented. In addition, several open questions are discussed.Comment: This is the author's version of the homonymous volume published by Springer. The final authenticated version is available online at: https://doi.org/10.1007/978-3-030-12564-6 Further updates and corrections may be made her

Topics in Nonstationary Time Series Analysis

Several interesting applications in areas such as neuroscience, economics, finance and seismology have led to the collection nonstationary time series data wherein the statistical properties of the observed process change across time. The analysis of nonstationary time series data is an important and challenging task with useful applications. In comparison to stationarity, modeling temporal dependence in nonstationary time series is more non-trivial, and numerous methods have been proposed to tackle this problem. Stationarity in time series is more coveted than nonstationarity and many of the existing techniques attempt to transform the problem of nonstationarity to a stationary time series setting. Change point detection is one such method that attempts to find time points wherein the statistical properties of the time series changed. We develop a nonparametric method to detect multiple change points in multivariate piecewise stationary processes when the locations and number of change points are unknown. Based on a test statistic that measures differences in the spectral density matrices through the L₂ norm, we sequentially identify points of local maxima in the test statistic and test for the significance of each of them being change points. In addition, the components responsible for the change in the covariance structure at each detected change point are identified. The asymptotic properties of the test for significant change points under the null and alternative hypothesis are derived. Another related method for handling nonstationarity is the recent technique of stationary subspace analysis (SSA) that aims at finding linear transformations of nonstationary processes that are stationary. We propose an SSA procedure for general multivariate second-order nonstationary processes. It relies on the asymptotic uncorrelatedness of the discrete Fourier transform of a stationary time series to define a measure of departure from stationarity; it is then minimized to find the stationary subspace. The dimension of the subspace is estimated using a sequential testing procedure and its asymptotic properties are discussed. We illustrate the broader applicability and better performance of our method in comparison to existing SSA methods through simulations and discuss an application in neuroeconomics. Here we apply our method to filter out noise in EEG brain signals from an economic choice task experiment. This improves prediction performance and more importantly reduces the number of trials needed from individuals in neuroeconomic experiments thereby aligning with the principle of simple and controlled designs in experimental and behavioral economics

Topics in Nonstationary Time Series Analysis

Several interesting applications in areas such as neuroscience, economics, finance and seismology have led to the collection nonstationary time series data wherein the statistical properties of the observed process change across time. The analysis of nonstationary time series data is an important and challenging task with useful applications. In comparison to stationarity, modeling temporal dependence in nonstationary time series is more non-trivial, and numerous methods have been proposed to tackle this problem. Stationarity in time series is more coveted than nonstationarity and many of the existing techniques attempt to transform the problem of nonstationarity to a stationary time series setting. Change point detection is one such method that attempts to find time points wherein the statistical properties of the time series changed. We develop a nonparametric method to detect multiple change points in multivariate piecewise stationary processes when the locations and number of change points are unknown. Based on a test statistic that measures differences in the spectral density matrices through the L₂ norm, we sequentially identify points of local maxima in the test statistic and test for the significance of each of them being change points. In addition, the components responsible for the change in the covariance structure at each detected change point are identified. The asymptotic properties of the test for significant change points under the null and alternative hypothesis are derived. Another related method for handling nonstationarity is the recent technique of stationary subspace analysis (SSA) that aims at finding linear transformations of nonstationary processes that are stationary. We propose an SSA procedure for general multivariate second-order nonstationary processes. It relies on the asymptotic uncorrelatedness of the discrete Fourier transform of a stationary time series to define a measure of departure from stationarity; it is then minimized to find the stationary subspace. The dimension of the subspace is estimated using a sequential testing procedure and its asymptotic properties are discussed. We illustrate the broader applicability and better performance of our method in comparison to existing SSA methods through simulations and discuss an application in neuroeconomics. Here we apply our method to filter out noise in EEG brain signals from an economic choice task experiment. This improves prediction performance and more importantly reduces the number of trials needed from individuals in neuroeconomic experiments thereby aligning with the principle of simple and controlled designs in experimental and behavioral economics

Random walks - a sequential approach

In this paper sequential monitoring schemes to detect nonparametric drifts are studied for the random walk case. The procedure is based on a kernel smoother. As a by-product we obtain the asymptotics of the Nadaraya-Watson estimator and its as- sociated sequential partial sum process under non-standard sampling. The asymptotic behavior differs substantially from the stationary situation, if there is a unit root (random walk component). To obtain meaningful asymptotic results we consider local nonpara- metric alternatives for the drift component. It turns out that the rate of convergence at which the drift vanishes determines whether the asymptotic properties of the monitoring procedure are determined by a deterministic or random function. Further, we provide a theoretical result about the optimal kernel for a given alternative

Developments in the Analysis of Spatial Data

Disregarding spatial dependence can invalidate methods for analyzingcross-sectional and panel data. We discuss ongoing work on developingmethods that allow for, test for, or estimate, spatial dependence. Muchof the stress is on nonparametric and semiparametric methods.

Pranab Kumar Sen: Life and works

In this article, we describe briefly the highlights and various accomplishments in the personal as well as the academic life of Professor Pranab Kumar Sen.Comment: Published in at http://dx.doi.org/10.1214/193940307000000013 the IMS Collections (http://www.imstat.org/publications/imscollections.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Testing temporal constancy of the spectral structure of a time series

Statistical inference for stochastic processes with time-varying spectral characteristics has received considerable attention in recent decades. We develop a nonparametric test for stationarity against the alternative of a smoothly time-varying spectral structure. The test is based on a comparison between the sample spectral density calculated locally on a moving window of data and a global spectral density estimator based on the whole stretch of observations. Asymptotic properties of the nonparametric estimators involved and of the test statistic under the null hypothesis of stationarity are derived. Power properties under the alternative of a time-varying spectral structure are discussed and the behavior of the test for fixed alternatives belonging to the locally stationary processes class is investigated.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ179 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm